Variations on the “exact factorization” theme

Xavier Gonze (), Jianqiang Sky Zhou and Lucia Reining
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Xavier Gonze: European Theoretical Spectroscopy Facility (ETSF) and Institute of Condensed Matter and Nanosciences, Université catholique de Louvain
Jianqiang Sky Zhou: European Theoretical Spectroscopy Facility (ETSF) and Sorbonne Université, CNRS, Institut des Nanosciences de Paris, UMR7588
Lucia Reining: European Theoretical Spectroscopy Facility (ETSF) and Laboratoire des Solides Irradiés, École Polytechnique, CNRS, CEA, Université Paris-Saclay

The European Physical Journal B: Condensed Matter and Complex Systems, 2018, vol. 91, issue 10, 1-21

Abstract: Abstract In a series of publications, Hardy Gross and co-workers have highlighted the interest of an “exact factorization” approach to the interacting electron-nuclei problem, be it time-independent or time-dependent. In this approach, an effective potential governs the dynamics of the nuclei such that the resulting N-body nuclear density is in principle exact. This contrasts with the more usual adiabatic approach, where the effective potential leads to an approximate nuclear density. Inspired by discussions with Hardy, we explore the factorization idea for arbitrary many-body Hamiltonians, generalizing the electron-nuclei case, with a focus on the static case. While the exact equations do not lead to any practical advantage, they are illuminating, and may therefore constitute a suitable starting point for approximations. In particular, we find that unitary transformations that diagonalize the coupling term for one of the sub-systems make exact factorization appealing. The algorithms by which the equations for the separate subsystems can be solved in the time-independent case are also explored. We illustrate our discussions using the two-site Holstein model and the quantum Rabi model. Two factorization schemes are possible: one where the boson field feels a potential determined by the electrons, and the reverse exact factorization, where the electrons feel a potential determined by the bosons; both are explored in this work. A comparison with a self-energy approach is also presented.

Date: 2018
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DOI: 10.1140/epjb/e2018-90278-2

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